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Coarse Graining Shannon and von Neumann Entropies

by Ana Alonso-Serrano 1,*,† and Matt Visser 2,†
Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, 18000 Prague, Czech Republic
School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editor: David Kubiznak
Entropy 2017, 19(5), 207;
Received: 4 April 2017 / Revised: 27 April 2017 / Accepted: 28 April 2017 / Published: 3 May 2017
(This article belongs to the Special Issue Black Hole Thermodynamics II)
The nature of coarse graining is intuitively “obvious”, but it is rather difficult to find explicit and calculable models of the coarse graining process (and the resulting entropy flow) discussed in the literature. What we would like to have at hand is some explicit and calculable process that takes an arbitrary system, with specified initial entropy S, and that monotonically and controllably drives the entropy to its maximum value. This does not have to be a physical process, in fact for some purposes it is better to deal with a gedanken-process, since then it is more obvious how the “hidden information” is hiding in the fine-grain correlations that one is simply agreeing not to look at. We shall present several simple mathematically well-defined and easy to work with conceptual models for coarse graining. We shall consider both the classical Shannon and quantum von Neumann entropies, including models based on quantum decoherence, and analyse the entropy flow in some detail. When coarse graining the quantum von Neumann entropy, we find it extremely useful to introduce an adaptation of Hawking’s super-scattering matrix. These explicit models that we shall construct allow us to quantify and keep clear track of the entropy that appears when coarse graining the system and the information that can be hidden in unobserved correlations (while not the focus of the current article, in the long run, these considerations are of interest when addressing the black hole information puzzle). View Full-Text
Keywords: coarse graining; entropy; information; Shannon entropy; von Neumann entropy coarse graining; entropy; information; Shannon entropy; von Neumann entropy
MDPI and ACS Style

Alonso-Serrano, A.; Visser, M. Coarse Graining Shannon and von Neumann Entropies. Entropy 2017, 19, 207.

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